Time and World-Lines

Let γ: [s1, s2] → M be a smooth, future-directed timelike curve in M with tangent field ξa. We associate with it an elapsed proper time (relative to gab) given by

∥γ∥= ∫s1s2 (gabξaξb)1/2 ds

This elapsed proper time is invariant under reparametrization of γ and is just what we would otherwise describe as the length of (the image of) γ . The following is another basic principle of relativity theory:

Clocks record the passage of elapsed proper time along their world-lines.

Again, a number of qualifications and comments are called for. We have taken for granted that we know what “clocks” are. We have assumed that they have worldlines (rather than worldtubes). And we have overlooked the fact that ordinary clocks (e.g., the alarm clock on the nightstand) do not do well at all when subjected to extreme acceleration, tidal forces, and so forth. (Try smashing the alarm clock against the wall.) Again, these concerns are important and raise interesting questions about the role of idealization in the formulation of physical theory. (One might construe an “ideal clock” as a point-size test object that perfectly records the passage of proper time along its worldline, and then take the above principle to assert that real clocks are, under appropriate conditions and to varying degrees of accuracy, approximately ideal.) But they do not have much to do with relativity theory as such. Similar concerns arise when one attempts to formulate corresponding principles about clock behavior within the framework of Newtonian theory.

Now suppose that one has determined the conformal structure of spacetime, say, by using light rays. Then one can use clocks, rather than free particles, to determine the conformal factor.

Let g′ab be a second smooth metric on M, with g′ab = Ω2gab. Further suppose that the two metrics assign the same lengths to timelike curves – i.e., ∥γ∥g′ab = ∥γ∥gab ∀ smooth, timelike curves γ: I → M. Then Ω = 1 everywhere. (Here ∥γ∥gab is the length of γ relative to gab.)

Let ξoa be an arbitrary timelike vector at an arbitrary point p in M. We can certainly find a smooth, timelike curve γ: [s1, s2] → M through p whose tangent at p is ξoa. By our hypothesis, ∥γ∥g′ab = ∥γ∥gab. So, if ξa is the tangent field to γ,

s1s2 (g’ab ξaξb)1/2 ds = ∫s1s2 (gabξaξb)1/2 ds

∀ s in [s1, s2]. It follows that g′abξaξb = gabξaξb at every point on the image of γ. In particular, it follows that (g′ab − gab) ξoa ξob = 0 at p. But ξoa was an arbitrary timelike vector at p. So, g′ab = gab at our arbitrary point p. The principle gives the whole story of relativistic clock behavior. In particular, it implies the path dependence of clock readings. If two clocks start at an event p and travel along different trajectories to an event q, then, in general, they will record different elapsed times for the trip. This is true no matter how similar the clocks are. (We may stipulate that they came off the same assembly line.) This is the case because, as the principle asserts, the elapsed time recorded by each of the clocks is just the length of the timelike curve it traverses from p to q and, in general, those lengths will be different.

Suppose we consider all future-directed timelike curves from p to q. It is natural to ask if there are any that minimize or maximize the recorded elapsed time between the events. The answer to the first question is “no.” Indeed, one then has the following proposition:

Let p and q be events in M such that p ≪ q. Then, for all ε > 0, there exists a smooth, future directed timelike curve γ from p to q with ∥γ ∥ < ε. (But there is no such curve with length 0, since all timelike curves have non-zero length.)

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If there is a smooth, timelike curve connecting p and q, there is also a jointed, zig-zag null curve connecting them. It has length 0. But we can approximate the jointed null curve arbitrarily closely with smooth timelike curves that swing back and forth. So (by the continuity of the length function), we should expect that, for all ε > 0, there is an approximating timelike curve that has length less than ε.

The answer to the second question (“Can one maximize recorded elapsed time between p and q?”) is “yes” if one restricts attention to local regions of spacetime. In the case of positive definite metrics, i.e., ones with signature of form (n, 0) – we know geodesics are locally shortest curves. The corresponding result for Lorentzian metrics is that timelike geodesics are locally longest curves.

Let γ: I → M be a smooth, future-directed, timelike curve. Then γ can be reparametrized so as to be a geodesic iff ∀ s ∈ I there exists an open set O containing γ(s) such that , ∀ s1, s2 ∈ I with s1 ≤ s ≤ s2, if the image of γ′ = γ|[s1, s2] is contained in O, then γ′ (and its reparametrizations) are longer than all other timelike curves in O from γ(s1) to γ(s2). (Here γ|[s1, s2] is the restriction of γ to the interval [s1, s2].)

Of all clocks passing locally from p to q, the one that will record the greatest elapsed time is the one that “falls freely” from p to q. To get a clock to read a smaller elapsed time than the maximal value, one will have to accelerate the clock. Now, acceleration requires fuel, and fuel is not free. So the above proposition has the consequence that (locally) “saving time costs money.” And proposition before that may be taken to imply that “with enough money one can save as much time as one wants.” The restriction here to local regions of spacetime is essential. The connection described between clock behavior and acceleration does not, in general, hold on a global scale. In some relativistic spacetimes, one can find future-directed timelike geodesics connecting two events that have different lengths, and so clocks following the curves will record different elapsed times between the events even though both are in a state of free fall. Furthermore – this follows from the preceding claim by continuity considerations alone – it can be the case that of two clocks passing between the events, the one that undergoes acceleration during the trip records a greater elapsed time than the one that remains in a state of free fall. (A rolled-up version of two-dimensional Minkowski spacetime provides a simple example)

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Two-dimensional Minkowski spacetime rolledup into a cylindrical spacetime. Three timelike curves are displayed: γ1 and γ3 are geodesics; γ2 is not; γ1 is longer than γ2; and γ2 is longer than γ3.

The connection we have been considering between clock behavior and acceleration was once thought to be paradoxical. Recall the so-called “clock paradox.” Suppose two clocks, A and B, pass from one event to another in a suitably small region of spacetime. Further suppose A does so in a state of free fall but B undergoes acceleration at some point along the way. Then, we know, A will record a greater elapsed time for the trip than B. This was thought paradoxical because it was believed that relativity theory denies the possibility of distinguishing “absolutely” between free-fall motion and accelerated motion. (If we are equally well entitled to think that it is clock B that is in a state of free fall and A that undergoes acceleration, then, by parity of reasoning, it should be B that records the greater elapsed time.) The resolution of the paradox, if one can call it that, is that relativity theory makes no such denial. The situations of A and B here are not symmetric. The distinction between accelerated motion and free fall makes every bit as much sense in relativity theory as it does in Newtonian physics.

A “timelike curve” should be understood to be a smooth, future-directed, timelike curve parametrized by elapsed proper time – i.e., by arc length. In that case, the tangent field ξa of the curve has unit length (ξaξa = 1). And if a particle happens to have the image of the curve as its worldline, then, at any point, ξa is called the particle’s four-velocity there.

Agamben and the Biopolitical – Nihilistic and Thanatopolitical Expressions. Thought of the Day 56.0

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Agamben’s logic of biopolitics as the logic of the symmetry between sovereign power and the sacredness of bare life should be understood in terms of its historico-ontological destiny. Although this theme is only hinted at in Homo Sacer and the volumes that follow it, Agamben resolutely maintains that biopolitics is inherently metaphysical. If on the one hand ‘the inclusion of bare life in the political realm constitutes the original […] nucleus of sovereign power’ and ‘biopolitics is at least as old as the sovereign exception’, on the other hand, this political nexus cannot be dissociated from the epochal situation of metaphysics. Here Agamben openly displays his Heideggerian legacy; bare life, that which in history is increasingly isolated by biopolitics as Western politics, must be strictly related to ‘pure being’, that which in history is increasingly isolated by Western metaphysics:

Politics [as biopolitics] appears as the truly fundamental structure of Western metaphysics insofar as it occupies the threshold on which the relation between the living being and the logos is realized. In the ‘politicization’ of bare life – the metaphysical task par excellence – the humanity of living man is decided.

Commentators have not as yet sufficiently emphasized how biopolitics is consequently nothing else than Agamben’s name for metaphysics as nihilism. More specifically, while bare life remains for him the ‘empty and indeterminate’ concept of Western politics – which is thus as such originally nihilistic – its forgetting goes together with the progressive coming to light of what it conceals. From this perspective, nihilism will therefore correspond to the modern and especially post-modern generalisation of the state of exception: ‘the nihilism in which we are living is […] nothing other than the coming to light of […] the sovereign relation as such’. In other words, nihilism reveals the paradox of the inclusive exclusion of bare life, homo sacer, qua foundation of sovereign power, as well as the fact that sovereign power cannot recognize itself for what it is. Beyond Foucault’s biopolitical thesis according to which modernity is increasingly characterized by the way in which power directly captures life as such as its object, what interests Agamben the most is:

the decisive fact that, together with the process by which exception everywhere becomes the rule, the realm of bare life – which is originally situated at the margins of the political order – gradually begins to coincide with the political realm.

The political is thus reduced to the biopolitical: the original repression of the sovereign relation on which Western politics has always relied is now inextricably bound up with its return in the guise of a radical biopoliticisation of the political. Like nihilism, such a generalisation of the state of exception – the fact that, today, we are all virtually homines sacri – is itself a profoundly ambiguous biopolitical phenomenon. Today’s state of exception both radicalizes – qualitatively and quantitatively – the thanatopolitical expressions of sovereignty (epitomized by the nazis’ extermination of the Jews for a mere ‘capacity to be killed’ inherent in their condition as such) and finally unmasks its hidden logic.

Agamben explicitly relates to the possibility of a ‘new politics’. Conversely, a new politics is unthinkable without an in-depth engagement with the historico-ontological dimension of sacratio and the structural political ambiguity of the state of exception. Although such new politics ‘remains largely to be invented’, very early on in Homo Sacer, Agamben unhesitatingly defines it as ‘a politics no longer founded on the exceptio of bare life’. beyond the exceptionalist logic – by now self-imploded – that unites sovereignty to bare life, Agamben seems to envisage a relaional politics that would succeed in ‘constructing the link between zoe and bios’. This link between the bare life of man and his political existence would ‘heal’ the original ‘fracture’ which is at the same time precisely what causes their progressive indistinction in the generalized state of exception. Having said this, Agamben also conceives of such new politics as a non-relational relation that ‘will […] have to put the very form of relation into question, and to ask if the political fact is not perhaps thinkable beyond relation and, thus, no longer in the form of a connection’.

The Politics of Speed and the Ascendancy of the Right. Thought of the Day 55.0

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Speed and Politics and (Popular Defense; Ecological Struggles) register the scope of the vast change in Virilio’s position. The problematic shifts from space to time, and from an expansive politics of mobilization and liberalization, to a defensive and conservative politics of resistance to acceleration and to a defence of the social. Responding to the appeals of theologians like Bonhoeffer, Virilio begins to warn of the dangers implied in the new state of the world, dangers to the experience of space of the city and of democracy, and of the new possibility of apocalypse brought about by new technologies and strategies available to and adopted by the military elite. In a sense the essay Speed and Politics, with its theory of power through control of movement a ‘dromocracy’, was the culmination of an analysis applicable to a world already passing away. If the proletariat still thinks in terms of the control of streets and physical movement, the military thinks otherwise: it thinks logistically in relation to new meeting points such as airports, highways and telecommunications. Communism died, fascism survives and has adapted. In this world, available in instantaneous communication and immediate information, a new permanent state of emergency is created which brings a sharp end to struggles in relative speed.

Virilio draws out these conclusions more dramatically in his book Popular Defense:

If… civilians could have resisted the assault of the war machine, gotten ahead of it, by creating a defence without a body, condensed nowhere, it is quite evident that today they don’t even realize that technology has surpassed this kind of defence.

This is because: ‘There is no need for an armed body to attack civilians, so long as the latter have been properly trained to turn on their radios or plug in their television sets’. In these conditions the political state declines, and where ‘hyper-communicability’ exists there grows totalitarian power. The right of armed defence by citizens is lost, while on the other hand ‘from now on’ the military power is so ‘shapeless’ it can no longer be identified as it installs itself in a regime of generalized security: an important and irreversible shift from a state of political and civil justice, to a state of logistical and military discipline. This is achieved through the systematic destruction of all the major forms of social solidarity which previously offered real resistance to the state: particularly the family, conceived by Virilio as essentially a combat unit. The liberation of women effectively weakens the solidarity of the family as a defensive form against the state. The resort to terrorism by ultra-left-wing groups again only serves to strengthen, not weaken, the war machine. This creates a paradox: the possibility that the revolution can succeed through control of the streets has been lost yet ‘there is no more revolution except in resistance’. Virilio returns to his bunker.

Black Hole Complementarity: The Case of the Infalling Observer

The four postulates of black hole complementarity are:

Postulate 1: The process of formation and evaporation of a black hole, as viewed by a distant observer, can be described entirely within the context of standard quantum theory. In particular, there exists a unitary S-matrix which describes the evolution from infalling matter to outgoing Hawking-like radiation.

Postulate 2: Outside the stretched horizon of a massive black hole, physics can be described to good approximation by a set of semi-classical field equations.

Postulate 3: To a distant observer, a black hole appears to be a quantum system with discrete energy levels. The dimension of the subspace of states describing a black hole of mass M is the exponential of the Bekenstein entropy S(M).

We take as implicit in postulate 2 that the semi-classical field equations are those of a low energy effective field theory with local Lorentz invariance. These postulates do not refer to the experience of an infalling observer, but states a ‘certainty,’ which for uniformity we label as a further postulate:

Postulate 4: A freely falling observer experiences nothing out of the ordinary when crossing the horizon.

To be more specific, we will assume that postulate 4 means both that any low-energy dynamics this observer can probe near his worldline is well-described by familiar Lorentz-invariant effective field theory and also that the probability for an infalling observer to encounter a quantum with energy E ≫ 1/rs (measured in the infalling frame) is suppressed by an exponentially decreasing adiabatic factor as predicted by quantum field theory in curved spacetime. We will argue that postulates 1, 2, and 4 are not consistent with one another for a sufficiently old black hole.

Consider a black hole that forms from collapse of some pure state and subsequently decays. Dividing the Hawking radiation into an early part and a late part, postulate 1 implies that the state of the Hawking radiation is pure,

|Ψ⟩= ∑ii⟩E ⊗|i⟩L —– (1)

Here we have taken an arbitrary complete basis |i⟩L for the late radiation. We use postulates 1, 2, and 3 to make the division after the Page time when the black hole has emitted half of its initial Bekenstein-Hawking entropy; we will refer to this as an ‘old’ black hole. The number of states in the early subspace will then be much larger than that in the late subspace and, as a result, for typical states |Ψ⟩ the reduced density matrix describing the late-time radiation is close to the identity. We can therefore construct operators acting on the early radiation, whose action on |Ψ⟩ is equal to that of a projection operator onto any given subspace of the late radiation.

To simplify the discussion, we treat gray-body factors by taking the transmission coefficients T to have unit magnitude for a few low partial waves and to vanish for higher partial waves. Since the total radiated energy is finite, this allows us to think of the Hawking radiation as defining a finite-dimensional Hilbert space.

Now, consider an outgoing Hawking mode in the later part of the radiation. We take this mode to be a localized packet with width of order rs corresponding to a superposition of frequencies O(r−1s). Note that postulate 2 allows us to assign a unique observer-independent s lowering operator b to this mode. We can project onto eigenspaces of the number operator bb. In other words, an observer making measurements on the early radiation can know the number of photons that will be present in a given mode of the late radiation.

Following postulate 2, we can now relate this Hawking mode to one at earlier times, as long as we stay outside the stretched horizon. The earlier mode is blue-shifted, and so may have frequency ω* much larger than O(r−1s) though still sub-Planckian.

Next consider an infalling observer and the associated set of infalling modes with lowering operators a. Hawking radiation arises precisely because

b = ∫0 dω B(ω)aω + C(ω)aω —– (2)

so that the full state cannot be both an a-vacuum (a|Ψ⟩ = 0) and a bb eigenstate. Here again we have used our simplified gray-body factors.

The application of postulates 1 and 2 has thus led to the conclusion that the infalling observer will encounter high-energy modes. Note that the infalling observer need not have actually made the measurement on the early radiation: to guarantee the presence of the high energy quanta it is enough that it is possible, just as shining light on a two-slit experiment destroys the fringes even if we do not observe the scattered light. Here we make the implicit assumption that the measurements of the infalling observer can be described in terms of an effective quantum field theory. Instead we could simply suppose that if he chooses to measure bb he finds the expected eigenvalue, while if he measures the noncommuting operator aa instead he finds the expected vanishing value. But this would be an extreme modification of the quantum mechanics of the observer, and does not seem plausible.

Figure below gives a pictorial summary of our argument, using ingoing Eddington-Finkelstein coordinates. The support of the mode b is shaded. At large distance it is a well-defined Hawking photon, in a predicted eigenstate of bb by postulate 1. The observer encounters it when its wavelength is much shorter: the field must be in the ground state aωaω = 0, by postulate 4, and so cannot be in an eigenstate of bb. But by postulate 2, the evolution of the mode outside the horizon is essentially free, so this is a contradiction.

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Figure: Eddington-Finkelstein coordinates, showing the infalling observer encountering the outgoing Hawking mode (shaded) at a time when its size is ω−1* ≪ rs. If the observer’s measurements are given by an eigenstate of aa, postulate 1 is violated; if they are given by an eigenstate of bb, postulate 4 is violated; if the result depends on when the observer falls in, postulate 2 is violated.

To restate our paradox in brief, the purity of the Hawking radiation implies that the late radiation is fully entangled with the early radiation, and the absence of drama for the infalling observer implies that it is fully entangled with the modes behind the horizon. This is tantamount to cloning. For example, it violates strong subadditivity of the entropy,

SAB + SBC ≥ SB + SABC —– (3)

Let A be the early Hawking modes, B be outgoing Hawking mode, and C be its interior partner mode. For an old black hole, the entropy is decreasing and so SAB < SA. The absence of infalling drama means that SBC = 0 and so SABC = SA. Subadditivity then gives SA ≥ SB + SA, which fails substantially since the density matrix for system B by itself is thermal.

Actually, assuming the Page argument, the inequality is violated even more strongly: for an old black hole the entropy decrease is maximal, SAB = SA − SB, so that we get from subadditivity that SA ≥ 2SB + SA.

Note that the measurement of Nb takes place entirely outside the horizon, while the measurement of Na (real excitations above the infalling vacuum) must involve a region that extends over both sides of the horizon. These are noncommuting measurements, but by measuring Nb the observer can infer something about what would have happened if Na had been measured instead. For an analogy, consider a set of identically prepared spins. If each is measured along the x-axis and found to be +1/2, we can infer that a measurement along the z-axis would have had equal probability to return +1/2 and −1/2. The multiple spins are needed to reduce statistical variance; similarly in our case the observer would need to measure several modes Nb to have confidence that he was actually entangled with the early radiation. One might ask if there could be a possible loophole in the argument: A physical observer will have a nonzero mass, and so the mass and entropy of the black hole will increase after he falls in. However, we may choose to consider a particular Hawking wavepacket which is already separated from the streched horizon by a finite amount when it is encountered by the infalling observer. Thus by postulate 2 the further evolution of this mode is semiclassical and not affected by the subsequent merging of the observer with the black hole. In making this argument we are also assuming that the dynamics of the stretched horizon is causal.

Thus far the asymptotically flat discussion applies to a black hole that is older than the Page time; we needed this in order to frame a sharp paradox using the entanglement with the Hawking radiation. However, we are discussing what should be intrinsic properties of the black hole, not dependent on its entanglement with some external system. After the black hole scrambling time, almost every small subsystem of the black hole is in an almost maximally mixed state. So if the degrees of freedom sampled by the infalling observer can be considered typical, then they are ‘old’ in an intrinsic sense. Our conclusions should then hold. If the black hole is a fast scrambler the scrambling time is rs ln(rs/lP), after which we have to expect either drama for the infalling observer or novel physics outside the black hole.

We note that the three postulates that are in conflict – purity of the Hawking radiation, absence of infalling drama, and semiclassical behavior outside the horizon — are widely held even by those who do not explicitly label them as ‘black hole complementarity.’ For example, one might imagine that if some tunneling process were to cause a shell of branes to appear at the horizon, an infalling observer would just go ‘splat,’ and of course Postulate 4 would not hold.

Noneism. Part 1.

Meinong

Noneism was created by Richard Routley. Its point of departure is the rejection of what Routley calls “The Ontological Assumption”. This assumption consists in the explicit or, more frequently, implicit belief that denoting always refers to existing objects. If the object, or objects, on which a proposition is about, do not exist, then these objects can only be one: the null entity. It is incredible that Frege believed that denoting descriptions without a real (empirical, theoretical, or ideal) referent denoted only the null set. And it is also difficult to believe that Russell sustained the thesis that non-existing objects cannot have properties and that propositions about these objects are false.

This means that we can have a very clear apprehension of imaginary objects, and quite clear intellection of abstract objects that are not real. This is possible because to determine an object we only need to describe it through its distinctive traits. This description is possible because an object is always chacterized through some definite notes. The amount of traits necessary to identify an object greatly varies. In some cases we need only a few, for instance, the golden mountain, or the blue bird; in other cases we need more, for instance, the goddess Venus or the centaur Chiron. In other instances the traits can be very numerous, even infinite. For instance the chiliedron, and the decimal number 0,0000…009, in which 9 comes after the first million zeros, have many traits. And the ordinal omega or any Hilbert space have infinite traits (although these traits can be reckoned through finite definitions). These examples show, in a convincing manner, that the Ontological Assumption is untenable. We must reject it and replace it with what Routley dubbs the Characterization Postulate. The Characterization Postulate says that, to be an object means to be characterized by determined traits. The set of the characterizing traits of an object can be called its “characteristic”. When the characteristic of an object is set up, the object is perfectly recognizable.

Once this postulate is adopted, its consequences are far reaching. Since we can characterize objects through any traits whatsoever, an object can not only be inexistent, it can even be absurd or inconsistent. For instance, the “squond” (the circle that is square and round). And we can make perfectly valid logical inferences from the premiss: x is the sqound:

(1) if x is the squond, then x is square
(2) if x is the squond, then x is round

So, the theory of objects has the widest realm of application. It is clear that the Ontological Assumption imposes unacceptable limits to logic. As a matter of fact, the existential quantifier of classical logic could not have been conceived without the Ontological Assumption. The expression “(∃x)Fx” means that there exists at least an object that has the property F (or, in extensional language, that there exists an x that is a member of the extension of F). For this reason, “∃x” is unappliable to non existing objects. Of course, in classical logic we can deny the existence of an Object, but we cannot say anything about Objects that have never existed and shall never exist (we are strictly speaking about classical logic). We cannot quantify individual variables of a first order predicate that do not refer to a real, actual, past or future entity. For instance, we cannot say “(∃x) (x is the eye of Polyphemus)”. This would be false, of course, because Polyphemus does not exist. But if the Ontological Assumption is set aside, it is true, within a mythological frame, that Polyphemus has a single eye and many other properties. And now we can understand why noneism leads to logical material-dependence.

As we have anticipated, there must be some limitations concerning the selection of the contradictory properties; otherwise the whole theory becomes inconsistent and is trivialized. To avoid trivialization neutral (noneist) logic distinguishes between two sorts of negation: the classical propositional negation: “8 is not P”, and the narrower negation: “8 is non-P”. In this way, and by applying some other technicalities (for instance, in case an universe is inconsistent, some kind of paraconsistent logic must be used) trivialization is avoided. With the former provisions, the Characterization Postulate can be applied to create inconsistent universes in which classical logic is not valid. For instance, a world in which there is a mysterious personage, that within determined but very subtle circumstances, is and is not at the same time in two different places. In this case the logic to be applied is, obviously, some kind of paraconsistent logic (the type to be selected depends on the characteristic of the personage). And in another universe there could be a jewel which has two false properties: it is false that it is transparent and it is false that it is opaque. In this kind of world we must use, clearly, some kind of paracomplete logic. To develop naive set theory (in Halmos sense), we must use some type of paraconsistent logic to cope with the paradoxes, that are produced through a natural way of mathematical reasoning; this logic can be of several orders, just like the classical. In other cases, we can use some kind of relevant and, a fortiori, paraconsistent logic; and so on, ad infinitum.

But if logic is content-dependent, and this dependence is a consequence of the Ontological Assumption’s rejection, what about ontology? Because the universes determined through the application of the Characterization Postulate may have no being (in fact, most of them do not), we cannot say that the objects that populate such universes are entities, because entities exist in the empirical world, or in the real world that underpins the phenomena, or (in a somewhat different way), in an ideal Platonic world. Instead of speaking about ontology, we should speak about objectology. In essence objectology is the discipline founded by Meinong (Theory of Objects), but enriched and made more precise by Routley and other noneist logicians. Its main division would be Ontology (the study of real physical and Platonic objects) and Medenology (the study of objects that have no existence).

Paradox of Phallocentrism. Thought of the Day 34.0

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The paradox of phallocentrism in aIl its manifestations is that it depends on the image of the castrated woman to give order and meaning to its world. An idea of woman stands as lynch pin to the system: it is her lack that produces the phallus as a symbolic presence, it is her desire to make good the lack that the phallus signifies. The function of woman in forming the patriarchal unconscious is two-fold. She first symbolises the castration threat by her real absence of a penis, and second thereby raises her child into the symbolic. Once this has been achieved, her meaning in the process is at an end, it does not last into the world of law and language except as a memory which oscillates between memory of maternal plenitude and memory of lack. Both are posited on nature (or on anatomy in Freud’s famous phrase). Woman’s desire is subjected to her image as bearer of the bleeding wound, she can exist only in relation to castration and cannot transcend it. She turns her child into the signifier of her own desire to possess a penis (the condition, she imagines, of entry into the symbolic). Either she must gracefully give way to the word, the Name of the Father and the Law, or else struggle to keep her child down with her in the half-light of the imaginary. Woman then stands in patriarchal culture as signifier for the male other, bound by a symbolic order in which man can live out his phantasies and obsessions through linguistic command by imposing them on the silent image of woman still tied to her place as bearer of meaning, not maker of meaning.

Biogrammatic Vir(Ac)tuality. Note Quote.

In Foucault’s most famous example, the prison acts as the confluence of content (prisoners) and expression (law, penal code) (Gilles Deleuze, Sean Hand-Foucault). Informal Diagrams are proliferate. As abstract machines they contain the transversal vectors that cut across a panoply of features (such as institutions, classes, persons, economic formation, etc), mapping from point to relational point, the generalized features of power economies. The disciplinary diagram explored by Foucault, imposes “a particular conduct upon a particular human multiplicity”. The imposition of force upon force affects and effectuates the felt experience of a life, a living. Deleuze has called the abstract machine “pure matter/function” in which relations between forces are nonetheless very real.

[…] the diagram acts as a non-unifying immanent cause that is co-extensive with the whole social field: the abstract machine is like the cause of the concrete assemblages that execute its relations; and these relations between forces take place ‘not above’ but within the very tissue of the assemblages they produce.

The processual conjunction of content and expression; the cutting edge of deterritorialization:

The relations of power and resistance between theory and practice resonate – becoming-form; diagrammatics as praxis, integrates and differentiates the immanent cause and quasi-cause of the actualized occasions of research/creation. What do we mean by immanent cause? It is a cause which is realized, integrated and distinguished in its effect. Or rather, the immanent cause is realized, integrated and distinguished by its effect. In this way there is a correlation or mutual presupposition between cause and effect, between abstract machine and concrete assemblages

Memory is the real name of the relation to oneself, or the affect of self by self […] Time becomes a subject because it is the folding of the outside…forces every present into forgetting but preserves the whole of the past within memory: forgetting is the impossibiltiy of return and memory is the necessity of renewal.

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The figure on the left is Henri Bergson’s diagram of an infinitely contracted past that directly intersects with the body at point S – a mobile, sensorimotor present where memory is closest to action. Plane P represents the actual present; plane of contact with objects. The AB segments represent repetitive compressions of memory. As memory contracts it gets closer to action. In it’s more expanded forms it is closer to dreams. The figure on the right extrapolates from Bergson’s memory model to describe the Biogrammatic ontological vector of the Diagram as it moves from abstract (informal) machine in the most expanded form “A” through the cone “tissue” to the phase-shifting (formal), arriving at the Strata of the P plane to become artefact. The ontological vector passes through the stratified, through the interval of difference created in the phase shift (the same phase shift that separates and folds content and expression to move vertically, transversally, back through to the abstract diagram.)

A spatio-temporal-material contracting-expanding of the abstract machine is the processual thinking-feeling-articulating of the diagram becoming-cartographic; synaesthetic conceptual mapping. A play of forces, a series of relays, affecting a tendency toward an inflection of the informal diagram becoming-form. The inflected diagram/biogram folds and unfolds perception, appearances; rides in the gap of becoming between content and expression; intuitively transduces the actualizing (thinking, drawing, marking, erasing) of matter-movement, of expressivity-movement. “To follow the flow of matter… is intuition in action.” A processual stage that prehends the process of the virtual actualizing;

the creative construction of a new reality. The biogrammatic stage of the diagrammatic is paradoxically double in that it is both the actualizing of the abstract machine (contraction) and the recursive counter-actualization of the formal diagram (détournement); virtual and actual.

It is the event-dimension of potential – that is the effective dimension of the interrelating of elements, of their belonging to each other. That belonging is a dynamic corporeal “abstraction” – the “drawing off” (transductive conversion) of the corporeal into its dynamism (yielding the event) […] In direct channeling. That is, in a directional channeling: ontological vector. The transductive conversion is an ontological vector that in-gathers a heterogeneity of substantial elements along with the already-constituted abstractions of language (“meaning”) and delivers them together to change. (Brian Massumi Parables for the Virtual Movement, Affect, Sensation)

Skin is the space of the body the BwO that is interior and exterior. Interstitial matter of the space of the body.

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The material markings and traces of a diagrammatic process, a ‘capturing’ becoming-form. A diagrammatic capturing involves a transductive process between a biogrammatic form of content and a form of expression. The formal diagram is thus an individuating phase-shift as Simondon would have it, always out-of-phase with itself. A becoming-form that inhabits the gap, the difference, between the wave phase of the biogrammatic that synaesthetically draws off the intermix of substance and language in the event-dimension and the drawing of wave phase in which partial capture is formalized. The phase shift difference never acquires a vectorial intention. A pre-decisive, pre-emptive drawing of phase-shifting with a “drawing off” the biogram.

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If effects realize something this is because the relations between forces or power relations, are merely virtual, potential, unstable vanishing and molecular, and define only possibilities of interaction so long as they do not enter a macroscopic whole capable of giving form to their fluid manner and diffuse function. But realization is equally an integration, a collection of progressive integrations that are initially local and then become or tend to become global, aligning, homogenizing and summarizing relations between forces: here law is the integration of illegalisms.

 

Dance of the Shiva, q’i (chee) and Tibetan Sunyata. Manifestation of Mysticism.

अनेजदेकं मनसो जवीयो नैनद्देवाप्नुवन्पूर्वमर्षत् ।
तद्धावतोऽन्यान्नत्येति तिष्ठत् तस्मिन्नापो मातरिश्वा दधाति ॥

anejadekaṃ manaso javīyo nainaddevāpnuvanpūrvamarṣat |
taddhāvato’nyānnatyeti tiṣṭhat tasminnāpo mātariśvā dadhāti ||

The self is one. It is unmoving: yet faster than the mind. Thus moving faster, It is beyond the reach of the senses. Ever steady, It outstrips all that run. By its mere presence, the cosmic energy is enabled to sustain the activities of living beings.

तस्मिन् मनसि ब्रह्मलोकादीन्द्रुतं गच्छति सति प्रथमप्राप्त इवात्मचैतन्याभासो गृह्यते अतः मनसो जवीयः इत्याह ।

tasmin manasi brahmalokādīndrutaṃ gacchati sati prathamaprāpta ivātmacaitanyābhāso gṛhyate ataḥ manaso javīyaḥ ityāha |

When the mind moves fast towards the farthest worlds such as the brahmaloka, it finds the Atman, of the nature of pure awareness, already there; hence the statement that It is faster than the mind.

नित्योऽनित्यानां चेतनश्चेतनानाम्
एको बहूनां यो विदधाति कामान् ।
तमात्मस्थं योऽनुपश्यन्ति धीराः
तेषां शान्तिः शाश्वतं नेतरेषाम् ॥

nityo’nityānāṃ cetanaścetanānām
eko bahūnāṃ yo vidadhāti kāmān |
tamātmasthaṃ yo’nupaśyanti dhīrāḥ
teṣāṃ śāntiḥ śāśvataṃ netareṣām ||

He is the eternal in the midst of non-eternals, the principle of intelligence in all that are intelligent. He is One, yet fulfils the desires of many. Those wise men who perceive Him as existing within their own self, to them eternal peace, and non else.

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Eastern mysticism approaches the manifestation of life in the cosmos and all that compose it from a position diametrically opposed to the view that prevailed until recently among the majority of Western scientists, philosophers, and religionists. Orientals see the universe as a whole, as an organism. For them all things are interconnected, links in a chain of beings permeated by consciousness which threads them together. This consciousness is the one life-force, originator of all the phenomena we know under the heading of nature, and it dwells within its emanations, urging them as a powerful inner drive to grow and evolve into ever more refined expressions of divinity. The One manifests, not only in all its emanations, but also through those emanations as channels: it is within them and yet remains transcendent as well.

The emphasis is on the Real as subject whereas in the West it is seen as object. If consciousness is the noumenal or subjective aspect of life in contrast to the phenomenal or objective — everything seen as separate objects — then only this consciousness can be experienced, and no amount of analysis can reveal the soul of Reality. To illustrate: for the ancient Egyptians, their numerous “gods” were aspects of the primal energy of the Divine Mind (Thoth) which, before the creation of our universe, rested, a potential in a subjective state within the “waters of Space.” It was through these gods that the qualities of divinity manifested.

A question still being debated runs: “How does the One become the many?” meaning: if there is a “God,” how do the universe and the many entities composing it come into being? This question does not arise among those who perceive the One to dwell in the many, and the many to live in the One from whom life and sustenance derive. Despite our Western separation of Creator and creation, and the corresponding distancing of “God” from human beings, Western mystics have held similar views to those of the East, e.g.: Meister Eckhart, the Dominican theologian and preacher, who was accused of blasphemy for daring to say that he had once experienced nearness to the “Godhead.” His friends and followers were living testimony to the charisma (using the word in its original connotation of spiritual magnetism) of those who live the life of love for fellow beings men like Johannes Tauler, Heinrich Suso, the “admirable Ruysbroeck,” who expressed views similar to those of Eastern exponents of the spiritual way or path.

In old China, the universe was described as appearing first as q’i (chee), an emanation of Light, not the physical light that we know, but its divine essence sometimes called Tien, Heaven, in contrast to Earth. The q’i energy polarized as Yang and Yin, positive and negative electromagnetism. From the action and interaction of these two sprang the “10,000 things”: the universe, our world, the myriads of beings and things as we perceive them to be. In other words, the ancient Chinese viewed our universe as one of process, the One energy, q’i, proliferating into the many.

In their paintings Chinese artists depict man as a small but necessary element in gigantic natural scenes. And since we are parts of the cosmos, we are embodiments of all its potentials and our relationship depends upon how we focus ourselves: (1) harmoniously, i.e., in accord with nature; or (2) disharmoniously, interfering with the course of nature. We therefore affect the rest: our environment, all other lives, and bear full responsibility for the outcome of our thoughts and acts, our motivations, our impacts. Their art students were taught to identify with what they were painting, because there is life in every thing, and it is this life with which they must identify, with boulders and rocks no less than with birds flying overhead. Matter, energy, space, are all manifestations of q’i and we, as parts thereof, are intimately connected with all the universe.

In India, the oneness of life was seen through the prism of successive manifestations of Brahman, a neuter or impersonal term in Sanskrit for divinity, the equivalent of what Eckhart called the Godhead. Brahman is the source of the creative power, Brahma, Eckhart’s Creator; and also the origin of the sustaining and supporting energy or Vishnu, and of the destructive/regenerative force or Siva. As these three operate through the cosmos, the “world” as we know it, so do they also through ourselves on a smaller scale according to our capacity. Matter is perceived to be condensed energy, Chit or consciousness itself. To quote from the Mundaka Upanishad:

By the energism of Consciousness Brahman is massed; from that: Matter is born and from Matter Life and Mind and the worlds . . .

In another Hindu scripture, it is stated that when Brahma awakened from his period of rest between manifestations, he desired to contemplate himself as he is. By gazing into the awakening matter particles as into a mirror, he stirred them to exhibit their latent divine qualities. Since this process involves a continuous unfoldment from the center within, an ever-becoming, there can never be an end to the creativity — universal “days” comprising trillions of our human years, followed by a like number of resting “nights.”

We feel within ourselves the same driving urge to grow that runs through the entire, widespread universe, to express more and more of what is locked up in the formless or subjective realm of Be-ness, awaiting the magic moment to come awake in our phase of life.

Tibetan metaphysics embraces all of this in discussing Sunyata, which can be viewed as Emptiness if we use only our outer senses, or as Fullness if we inwardly perceive it to be full of energies of limitless ranges of wave-lengths/frequencies. This latter aspect of Space is the great mother of all, ever fecund, from whose “heart” emerge endless varieties of beings, endless forces, ever-changing variations — like the pulsing energies the new physicists perceive nuclear subparticles to be.

In the Preface to his Tao of physics Fritjof Capra tells how one summer afternoon he had a transforming experience by the seashore as he watched the waves rolling in and felt the rhythm of his own breathing. He saw dancing motes revealed in a beam of sunlight; particles of energy vibrating as molecules and atoms; cascades of energy pouring down upon us from outer space. All of this coming and going, appearing and disappearing, he equated with the Indian concept of the dance of Siva . . . he felt its rhythm, “heard” its sound, and knew himself to be a part of it. Through this highly personal, indeed mystical, experience Capra became aware of his “whole environment as being engaged in a gigantic cosmic dance.”

This is the gist of the old Chinese approach to physics: students were taught gravitation by observing the petals of a flower as they fall gracefully to the ground. As Gary Zukav expresses it in his Dancing Wu Li Masters: An Overview of the New Physics:

The world of particle physics is a world of sparkling energy forever dancing with itself in the form of its particles as they twinkle in and out of existence, collide, transmute, and disappear again.

That is: the dance of Siva is the dance of attraction and repulsion between charged particles of the electromagnetic force. This is a kind of “transcendental” physics, going beyond the “world of opposites” and approaching a mystical view of the larger Reality that is to our perceptions an invisible foundation of what we call “physical reality.” It is so far beyond the capacity or vocabulary of the mechanically rational part of our mind to define, that the profound Hindu scripture Isa Upanishad prefers to suggest the thought by a paradox:

तदेजति तन्नैजति तद्दूरे तद्वन्तिके ।
तदन्तरस्य सर्वस्य तदु सर्वस्यास्य बाह्यतः ॥

tadejati tannaijati taddūre tadvantike |
tadantarasya sarvasya tadu sarvasyāsya bāhyataḥ ||

It moves. It moves not.  It is far, and it is near. It is within all this, And It is verily outside of all this.

Indeed, there is a growing recognition mostly by younger physicists that consciousness is more than another word for awareness, more than a by-product of cellular activity (or of atomic or subatomic vibrations). For instance, Jack Sarfatti, a quantum physicist, says that signals pulsating through space provide instant communication between all parts of the cosmos. “These signals can be likened to pulses of nerve cells of a great cosmic brain that permeates all parts of space (Michael Talbot, Mysticism and the New Physics).” Michael Talbot quotes Sir James Jeans’ remark, “the universe is more like a giant thought than a giant machine,” commenting that the “substance of the great thought is consciousness” which pervades all space. Or as Schrödinger would have it:

Consciouness is never experienced in the plural, only in the singular….Consciouness is a singular of which the plural is unknown; that; there is only one thing and that, what seems to be a plurality is merely a series of different aspects of this one thing, produced by a deception (the Indian Maya).

Other phenomena reported as occurring in the cosmos at great distances from each other, yet simultaneously, appear to be connected in some way so far unexplained, but to which the term consciousness has been applied.

In short, the mystic deals with direct experience; the intuitive scientist is open-minded, and indeed the great discoveries such as Einstein’s were made by amateurs in their field untrammeled by prior definitions and the limitations inherited from past speculations. This freedom enabled them to strike out on new paths that they cleared and paved. The rationalist tries to grapple with the problems of a living universe using only analysis and whatever the computer functions of the mind can put together.

The theosophic perspective upon universal phenomena is based on the concept of the ensoulment of the cosmos. That is: from the smallest subparticle we know anything about to the largest star-system that has been observed, each and all possess at their core vitality, energy, an active something propelling towards growth, evolution of faculties from within.

The only “permanent” in the whole universe is motion: unceasing movement, and the ideal perception is a blend of the mystical with the scientific, the intuitive with the rational.

Typicality. Cosmological Constant and Boltzmann Brains. Note Quote.

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In a multiverse we would expect there to be relatively many universe domains with large values of the cosmological constant, but none of these allow gravitationally bound structures (such as our galaxy) to occur, so the likelihood of observing ourselves to be in one is essentially zero.

The cosmological constant has negative pressure, but positive energy.  The negative pressure ensures that as the volume expands then matter loses energy (photons get red shifted, particles slow down); this loss of energy by matter causes the expansion to slow down – but the increase in energy of the increased volume is more important .  The increase of energy associated with the extra space the cosmological constant fills has to be balanced by a decrease in the gravitational energy of the expansion – and this expansion energy is negative, allowing the universe to carry on expanding.  If you put all the terms on one side in the Friedmann equation – which is just an energy balancing equation – (with the other side equal to zero) you will see that the expansion energy is negative, whereas the cosmological constant and matter (including dark matter) all have positive energy.

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However, as the cosmological constant is decreased, we eventually reach a transition point where it becomes just small enough for gravitational structures to occur. Reduce it a bit further still, and you now get universes resembling ours. Given the increased likelihood of observing such a universe, the chances of our universe being one of these will be near its peak. Theoretical physicist Steven Weinberg used this reasoning to correctly predict the order of magnitude of the cosmological constant well before the acceleration of our universe was even measured.

Unfortunately this argument runs into conceptually murky water. The multiverse is infinite and it is not clear whether we can calculate the odds for anything to happen in an infinite volume of space- time. All we have is the single case of our apparently small but positive value of the cosmological constant, so it’s hard to see how we could ever test whether or not Weinberg’s prediction was a lucky coincidence. Such questions concerning infinity, and what one can reasonably infer from a single data point, are just the tip of the philosophical iceberg that cosmologists face.

Another conundrum is where the laws of physics come from. Even if these laws vary across the multiverse, there must be, so it seems, meta-laws that dictate the manner in which they are distributed. How can we, inhabitants on a planet in a solar system in a galaxy, meaningfully debate the origin of the laws of physics as well as the origins of something, the very universe, that we are part of? What about the parts of space-time we can never see? These regions could infinitely outnumber our visible patch. The laws of physics could differ there, for all we know.

We cannot settle any of these questions by experiment, and this is where philosophers enter the debate. Central to this is the so-called observational-selection effect, whereby an observation is influenced by the observer’s “telescope”, whatever form that may take. But what exactly is it to be an observer, or more specifically a “typical” observer, in a system where every possible sort of observer will come about infinitely many times? The same basic question, centred on the role of observers, is as fundamental to the science of the indefinitely large (cosmology) as it is to that of the infinitesimally small (quantum theory).

This key issue of typicality also confronted Austrian physicist and philosopher Ludwig Boltzmann. In 1897 he posited an infinite space-time as a means to explain how extraordinarily well-ordered the universe is compared with the state of high entropy (or disorder) predicted by thermodynamics. Given such an arena, where every conceivable combination of particle position and momenta would exist somewhere, he suggested that the orderliness around us might be that of an incredibly rare fluctuation within an infinite space-time.

But Boltzmann’s reasoning was undermined by another, more absurd, conclusion. Rare fluctuations could also give rise to single momentary brains – self aware entities that spontaneously arises through random collisions of particles. Such “Boltzmann brains”, the argument goes, are far more likely to arise than the entire visible universe or even the solar system. Ludwig Boltzmann reasoned that brains and other complex, orderly objects on Earth were the result of random fluctuations. But why, then, do we see billions of other complex, orderly objects all around us? Why aren’t we like the lone being in the sea of nonsense?Boltzmann theorized that if random fluctuations create brains like ours, there should be Boltzmann brains floating around in space or sitting alone on uninhabited planets untold lightyears away. And in fact, those Boltzmann brains should be incredibly more common than the herds of complex, orderly objects we see here on Earth. So we have another paradox. If the only requirement of consciousness is a brain like the one in your head, why aren’t you a Boltzmann brain? If you were assigned to experience a random consciousness, you should almost certainly find yourself alone in the depths of space rather than surrounded by similar consciousnesses. The easy answers seem to all require a touch of magic. Perhaps consciousness doesn’t arise naturally from a brain like yours but requires some metaphysical endowment. Or maybe we’re not random fluctuations in a thermodynamic soup, and we were put here by an intelligent being. An infinity of space would therefore contain an infinitude of such disembodied brains, which would then be the “typical observer”, not us. OR. Starting at the very beginning: entropy must always stay the same or increase over time, according to the second law of thermodynamics. However, Boltzmann (the Ludwig one, not the brain one) formulated a version of the law of entropy that was statistical. What this means for what you’re asking is that while entropy almost always increases or stays the same, over billions of billions of billions of billions of billions…you get the idea years, entropy might go down a bit. This is called a fluctuation. So backing up a tad, if entropy always increases/stays the same, what is surprising for cosmologists is that the universe started in such a low-entropy state. So to (try) to explain this, Boltzmann said, hey, what if there’s a bigger universe that our universe is in, and it is in a state of the most possible entropy, or thermal equilibrium. Then, let’s say it exists for a long long time, those billions we talked about earlier. There’ll be statistical fluctuations, right? And those statistical fluctuations might be represented by the birth of universes. Ahem, our universe is one of them. So now, we get into the brains. Our universe must be a HUGE statistical fluctuation comparatively to other fluctuations. I mean, think about it. If it is so nuts for entropy to decrease by just a little tiny bit, how nuts would it be for it to decrease enough for the birth of a universe to happen!? So the question is, why aren’t we just brains? That is, why aren’t we a statistical fluctuation just big enough for intelligent life to develop, look around, see it exists, and melt back into goop. And it is this goopy-not-long-existing intelligent life that is a Boltzmann brain. This is a huge challenge to the Boltzmann (Ludwig) theory.

Can this bizarre vision possibly be real, or does it indicate something fundamentally wrong with our notion of “typicality”? Or is our notion of “the observer” flawed – can thermodynamic fluctuations that give rise to Boltzmann’s brains really suffice? Or could a futuristic supercomputer even play the Matrix-like role of a multitude of observers?

Paulian Idea, Bayesianism and Quantum Solipsism. Note Quote.

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The best way to begin a more thoroughly QBist delineation of quantum mechanics is to start with two choice quotes on personalist Bayesianism itself. The first is from Hampton, Moore, and Thomas,

Bruno de Finetti believes there is no need to assume that the probability of some event has a uniquely determinable value. His philosophical view of probability is that it expresses the feeling of an individual and cannot have meaning except in relation to him.

and the second from Dennis Lindley,

The Bayesian, subjectivist, or coherent, paradigm is egocentric. It is a tale of one person contemplating the world and not wishing to be stupid (technically, incoherent). He realizes that to do this his statements of uncertainty must be probabilistic.

These two quotes make it absolutely clear that personalist Bayesianism is a “single-user theory.” Thus, QBism must inherit at least this much egocentrism in its view of quantum states ρ.

For, the “Paulian Idea”—which is also essential to the QBist view—goes further still. It says that the outcomes to quantum measurements are single-user as well! That is to say, when an agent writes down her degrees of belief for the outcomes of a quantum measurement, what she is writing down are her degrees of belief about her potential personal experiences arising in consequence of her actions upon the external world

With regard to the Paulian Idea there are two points that are decisive for dismissing the charge of solipsism. One is the conceptual split of the world into two parts—one an agent and the other an external quantum system—that gets the discussion of quantum measurement off the ground in the first place. If such a split were not needed for making sense of the question of actions (actions upon what? in what? with respect to what?), it simply would not have been made. Imagining a quantum measurement without an autonomous quantum system participating in the process would be as paradoxical as the Zen koan of the sound of a single hand clapping. The second point is that once the agent chooses an action {Ei}, the particular consequence Ek of it is beyond his control. That is to say, the particular outcome of a quantum measurement is not a product of his desires, whims, or fancies—this is the very reason he uses the calculus of probabilities in the first place: they quantify his uncertainty, an uncertainty that, try as he might, he cannot get around. So, implicit in this whole picture—this whole Paulian Idea—is an “external world . . . made of something,” just as Martin Gardner(1) calls for. It is only that quantum theory is a rather small theory: Its boundaries are set by being a handbook for agents immersed within that “world made of something.”

(1) “Well then, it is incomplete after all. Go seek hidden variables!” But that is to misunderstand the problematic here. Theories of decision that really are theories of decision just don’t “port” to theories or visions of the world in that way. From the point of view of being a theory for taking actions and gambles, quantum theory is already all that it can be.